Digital Image Processing Image Enhancement (Spatial Filtering 2)

Contents In this lecture we will look at more spatial filtering techniques Spatial filtering refresher Sharpening filters 1st derivative filters 2nd derivative filters Combining filtering techniques

Spatial Filtering Refresher Origin x a b c d e f g h i r s t u v w x y z * Original Image Pixels Filter Simple 3*3 Neighbourhood e 3*3 Filter eprocessed = v*e + r*a + s*b + t*c + u*d + w*f + x*g + y*h + z*i y Image f (x, y) The above is repeated for every pixel in the original image to generate the smoothed image

Sharpening Spatial Filters Previously we have looked at smoothing filters which remove fine detail Sharpening spatial filters seek to highlight fine detail Remove blurring from images Highlight edges Sharpening filters are based on spatial differentiation

Spatial Differentiation Differentiation measures the rate of change of a function Let’s consider a simple 1 dimensional example Images taken from Gonzalez & Woods, Digital Image Processing (2002)

Spatial Differentiation Images taken from Gonzalez & Woods, Digital Image Processing (2002) A B

1st Derivative The formula for the 1st derivative of a function is as follows: It’s just the difference between subsequent values and measures the rate of change of the function

1st Derivative (cont…) 5 4 3 2 1 6 7 -1 6 -6 1 2 -2 7

2nd Derivative The formula for the 2nd derivative of a function is as follows: Simply takes into account the values both before and after the current value

2nd Derivative (cont…) 5 4 3 2 1 6 7 -1 1 6 -12 -4 7 -7

Using Second Derivatives For Image Enhancement The 2nd derivative is more useful for image enhancement than the 1st derivative Stronger response to fine detail Simpler implementation We will come back to the 1st order derivative later on The first sharpening filter we will look at is the Laplacian Isotropic One of the simplest sharpening filters We will look at a digital implementation

The Laplacian The Laplacian is defined as follows: where the partial 1st order derivative in the x direction is defined as follows: and in the y direction as follows:

The Laplacian (cont…) So, the Laplacian can be given as follows: We can easily build a filter based on this 1 -4

The Laplacian (cont…) Applying the Laplacian to an image we get a new image that highlights edges and other discontinuities Images taken from Gonzalez & Woods, Digital Image Processing (2002) Original Image Laplacian Filtered Image Laplacian Filtered Image Scaled for Display

But That Is Not Very Enhanced! The result of a Laplacian filtering is not an enhanced image We have to do more work in order to get our final image Subtract the Laplacian result from the original image to generate our final sharpened enhanced image Laplacian Filtered Image Scaled for Display Images taken from Gonzalez & Woods, Digital Image Processing (2002)

Laplacian Image Enhancement Images taken from Gonzalez & Woods, Digital Image Processing (2002) - = Original Image Laplacian Filtered Image Sharpened Image In the final sharpened image edges and fine detail are much more obvious

Laplacian Image Enhancement Images taken from Gonzalez & Woods, Digital Image Processing (2002)

Simplified Image Enhancement The entire enhancement can be combined into a single filtering operation

Simplified Image Enhancement (cont…) This gives us a new filter which does the whole job for us in one step Images taken from Gonzalez & Woods, Digital Image Processing (2002) -1 5

Simplified Image Enhancement (cont…) Images taken from Gonzalez & Woods, Digital Image Processing (2002)

Variants On The Simple Laplacian There are lots of slightly different versions of the Laplacian that can be used: Images taken from Gonzalez & Woods, Digital Image Processing (2002) 1 -4 1 -8 Simple Laplacian Variant of Laplacian -1 9

Simple Convolution Tool In Java A great tool for testing out different filters From the book “Image Processing tools in Java” Available from webCT later on today To launch: java ConvolutionTool Moon.jpg

1st Derivative Filtering Implementing 1st derivative filters is difficult in practice For a function f(x, y) the gradient of f at coordinates (x, y) is given as the column vector:

1st Derivative Filtering (cont…) The magnitude of this vector is given by: For practical reasons this can be simplified as:

1st Derivative Filtering (cont…) There is some debate as to how best to calculate these gradients but we will use: which is based on these coordinates z1 z2 z3 z4 z5 z6 z7 z8 z9

Sobel Operators Based on the previous equations we can derive the Sobel Operators To filter an image it is filtered using both operators the results of which are added together -1 -2 1 2 -1 1 -2 2

Sobel Example Sobel filters are typically used for edge detection Images taken from Gonzalez & Woods, Digital Image Processing (2002) An image of a contact lens which is enhanced in order to make defects (at four and five o’clock in the image) more obvious

1st & 2nd Derivatives Comparing the 1st and 2nd derivatives we can conclude the following: 1st order derivatives generally produce thicker edges 2nd order derivatives have a stronger response to fine detail e.g. thin lines 1st order derivatives have stronger response to grey level step 2nd order derivatives produce a double response at step changes in grey level

Summary In this lecture we looked at: Sharpening filters 1st derivative filters 2nd derivative filters Combining filtering techniques

Combining Spatial Enhancement Methods Successful image enhancement is typically not achieved using a single operation Rather we combine a range of techniques in order to achieve a final result This example will focus on enhancing the bone scan to the right Images taken from Gonzalez & Woods, Digital Image Processing (2002)

Combining Spatial Enhancement Methods (cont…) Images taken from Gonzalez & Woods, Digital Image Processing (2002) (a) Laplacian filter of bone scan (a) (b) Sharpened version of bone scan achieved by subtracting (a) and (b) (c) Sobel filter of bone scan (a) (d)

Combining Spatial Enhancement Methods (cont…) Result of applying a power-law trans. to (g) (h) Images taken from Gonzalez & Woods, Digital Image Processing (2002) Sharpened image which is sum of (a) and (f) (g) The product of (c) and (e) which will be used as a mask (f) (e) Image (d) smoothed with a 5*5 averaging filter

Combining Spatial Enhancement Methods (cont…) Compare the original and final images Images taken from Gonzalez & Woods, Digital Image Processing (2002)